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### HW-14s

Course: PHYSICS 112, Summer 2009
School: Iowa State
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Word Count: 451

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1. SOLUTIONS A diffraction grating has lines spaced 2.00 .m apart. Let's consider white visible light (wavelengths ranging from 0.400 .m for violet light to 0.700 .m for red light) shining on it. (a) Determine all the angles for which interference maxima occur for violet light. sin ) oe 7-. oe 70.400 .m2.00 .m oe 70.200 7 oe 0 sin )0 oe 0 )0 oe 0 7 oe 1 sin )1 oe 0.200 )0 oe 11.5 7 oe 2 sin )2 oe 0.400 )0 oe...

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1. SOLUTIONS A diffraction grating has lines spaced 2.00 .m apart. Let's consider white visible light (wavelengths ranging from 0.400 .m for violet light to 0.700 .m for red light) shining on it. (a) Determine all the angles for which interference maxima occur for violet light. sin ) oe 7-. oe 70.400 .m2.00 .m oe 70.200 7 oe 0 sin )0 oe 0 )0 oe 0 7 oe 1 sin )1 oe 0.200 )0 oe 11.5 7 oe 2 sin )2 oe 0.400 )0 oe 23.6 7 oe 3 sin )3 oe 0.600 )0 oe 36.9 7 oe 4 sin )4 oe 0.800 )0 oe 53.1 7 oe 5 sin )5 oe 1.000 )0 oe 90 (b) Determine all the angles for which interference maxima occur for red light. sin ) oe 7-. oe 70.700 .m2.00 .m oe 70.350 7 oe 0 sin )0 oe 0 )0 oe 0 7 oe 1 sin )1 oe 0.350 )0 oe 20.5 7 oe 2 sin )2 oe 0.700 )0 oe 44.4 7 oe 3 sin )3 oe 1.050 which is impossible (c) Over what range of angles are the first, second, etc. orders of the spectrum found? First order: 11.5 to 20.5 Second order: 23.6 to 44.4 Third order (partial) begins at 36.9 Fourth order (partial) begins at 53.1 (d) Which orders are complete (i.e., have angles less than 90 for all wavelengths)? Only the first and second orders are complete. (e) Which orders are completely separate, and do not overlap with any other orders? Only the first order is completely separate. The second order overlaps with the partial third order spectrum. 2. diffraction A pattern for monochromatic light of wavelength - oe 500 nm is observed from a single slit of width + oe 0.08 mm. How would the pattern be affected by: Doubling the slit width: The pattern will tighten up (narrow). Halving the slit width: The pattern will spread out. Changing the light to blue light: - is smaller so the pattern will tighten up (narrow). Changing the light to red light: - is larger so the pattern will spread out. 3. It is generally considered undesirable to "close down" a camera lens to a very small aperture, because the quality of the picture is adversely affected. Why? Too small an opening leads to a lot of diffraction of the light passing through the lens opening, and this degrades the image. 4. What is the angle corresponding to the first diffraction minimum for sounds of human speech (say 1000 Hz) passing through a doorway 1.0 m wide? We can use sin ) oe -/+ using the wavelength of the sound (- oe to find sin ) oe 0.34 m 1.00 m @ 0 oe 340 m/s 1000 Hz oe 0.34 m) oe 0.34, or ) oe 19.9. Express in words the meaning of your result. Since the diffraction angle is large, it is clear that it is not necessary to be directly on the other side of the door to hear the sounds; they will easily diffract out at large angles. Of course, they also reflect around easily.
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Iowa State - PHYSICS - 112
HOMEWORK FOR TUESDAY, AUGUST 2, 20051. Consider the atom denoted by 183 Au 79 The common name of this element is _. It has _ electrons, _ protons, _ neutrons, and _ quarks. This atom is radioactive and decays by alpha decay with a half-life of 49 seconds
Iowa State - PHYSICS - 112
1. Consider the atom denoted by 183 Au 79 The common name of this element is: gold It has 79 electrons, 79 protons, and 183 79 oe 104 neutrons. This atom is radioactive and decays by alpha decay with a half-life of 49 seconds. Determine the fraction of it
Iowa State - PHYSICS - 112
REFRACTION WORKSHEET &quot;Triangle&quot; Diagram: On other pages in this set you will find a large triangle that represents a triangular prism. We want to follow the path of a light ray striking one of the surfaces as it passes through the prism and exits one of t
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 1ELECTRIC CHARGES Electric charges are positive, negative, or zero; they are measured in coulombs (symbol: C). Electric charges are quantized: every object in the universe has a electric charge ; oe , 8/, where 8 oe 0, 1, 2,
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 2CURRENT M rate at which electic charge is flowing in a wire M oe ?UX RESISTANCE V and POTENTIAL DIFFERENCE ?Z related by Ohm's law: ?Z oe MV . The power dissipated in a resistor has magnitude T oe M ?Z oe M 2 V oe ?Z 2 V .
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 3t MAGNETIC FLUX: The magnetic flux through a plane circuit is F oe FE cos ), where ) is the angle between F 2 and the normal ( perpendicular) to the plane of the circuit. Units: T-m . MAGNETIC INDUCTION The current induced
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2003 - EXAM 1 - February 12, 2003 Name: _ Recitation Section Number: _ SHOW YOUR WORK! Although some of these problems are multiple-choice, full credit will be given only if you explain how you arrived at your answer. Either show your
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2003 - EXAM 3 SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially in a calculation) or give a short explanation. Nothing elaborate is required, but the grader
Iowa State - PHYSICS - 112
PHYSICS 112 - EXAM 4 (first part of final exam) - SPRING 2003Use only a right-handed coordinate system ( B C D along thumb, forefinger, and middle finger of your right hand when these three fingers are perpendicular to one another). Show your calculation
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2004 - EXAM 2 NO CALCULATORS. SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially in a calculation) or give a short explanation. Nothing elaborate is required
Iowa State - PHYSICS - 112
PHYSICS 112 - FINAL EXAM (second, comprehensive part) - SPRING 2005cos2 0 oe 1 cos2 30 oe 0.75 cos2 45 oe 0.50 cos2 60 oe 0.25 cos2 90 oe 0.1. [9 points total] An electric charge of 2.0 .C is placed at a point 3.0 m from a fixed electric charge of 3.0 .
Iowa State - PHYSICS - 112
PHYSICS 112 - SUMMER 2005 - EXAM 3 Name: _ Recitation Section: _NO CALCULATORS ALLOWED - BUT, LUCKILY, NO CALCULATORS NEEDED. SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially
Iowa State - PHYSICS - 112
USEFUL INFORMATIONIMPORTANT PHYSICAL CONSTANTS - 299,792,458 m/s 300 , 10) m/s K oe 6.673 , 10&quot; N m# /kg# 5/ oe1 41%0(speed of light) (gravitational constant) (electrical constant) (Planck's constant) (permittivity of free space) (fundamental electric
Iowa State - PHYSICS - 112
TOPIC 1. REVIEWINTRODUCTIONIn Physics 112 we will be making heavy use of vectors, so we will start with a review of vectors. The important vector quantities we will be discussing are electric forces, electric fields, electric currents, magnetic forces,
Iowa State - PHYSICS - 112
TOPIC 2. ELECTRICITY Topic 2A. Electric Charges and ForcesELECTRIC CHARGEObjects, including elementary particles like the electron and proton, have a number of properties. The one we have studied the most so far is mass. The mass of an object is importa
Iowa State - PHYSICS - 112
TOPIC 3. Electric Currents ELECTRIC CURRENTSWhat causes charges to flow, and what hinders the free flow of charge? The most important practical applications of electrical phenomena are in the innumerable forms of electronic devices. In these, it is the m
Iowa State - PHYSICS - 112
Topic 4. Magnetic Forces and FieldsDo currents exert forces on each other in the same way that charges do? Our study of electric forces and electric fields began with a simple experimental observation: charged particles exert forces on each other. From t
Iowa State - STAT - 511
IntroductionOne possible model: yij = i +ijIntroduction (continued)The first part of Stat 511 re-examines methods from Stat 500 from a linear models perspective.A simple study: does a proprietary food additive increase milk production in dairy cows?
Iowa State - STAT - 511
IntroductionThe first part of Stat 511 re-examines methods from Stat 500 from a linear models perspective. A simple study: does a proprietary food additive increase milk production in dairy cows? 6 cows, housed one per stall. Randomly choose 3 to get foo
Iowa State - STAT - 511
Geometry of the Gauss-Markov Linear ModelX is a linear combination of the columns of X: 1 . X = [x1 , . . . , xp ] . = 1 x1 + + p xp . . p The set of all possible linear combinations of the columns of X is called the column space of X and is denoted by C
Iowa State - STAT - 511
Geometry of the Gauss-Markov Linear ModelReminder from the last section of the notes: y = X + We saw two possible X matrices for the t-test. This section focuses on the Q: does it matter which X we use? Important pieces of information for what follows: X
Iowa State - STAT - 511
Estimating Estimable Functions of In the Gauss-Markov or Normal Theory Gauss-Markov Linear Model, the distribution of y depends on only through X, i.e., y (X, 2 I) or y N(X, 2 I)The Response Depends on Only through XWe now shift attention from E(y) to
Iowa State - STAT - 511
Estimating Estimable Functions of We now shift attention from E(y) to the parameter vector . Remember our t-test questions about 1 - 2 and 1 - 2 ? y y Those are questions about or linear combinations of . We've seen some models where there is a unique so
Iowa State - STAT - 511
Proof of the Gauss-Markov TheoremSuppose dy is any linear unbiased estimator other than the OLS ^ estimator C. ^ Need to show Var(dy) &gt; Var(C). ^ ^ Can relate the two Var by writing Var(dy) = Var(dy - C + C) ^ ^ Var(dy) = Var(dy - C + C) ^ ^ ^ ^ = Var(dy
Iowa State - STAT - 511
Proof of the Gauss-Markov TheoremGauss-Markov Th'm: ^ The OLS estimator, C, is the unique BLUE of C in GM model: y = X + , N(0, 2 I) ^ Need to show Var(C) is strictly less than the variance of any other linear unbiased estimator of C for all IRp and 2 IR
Iowa State - STAT - 511
Estimating Estimable Functions of : 11 12 22 23 33 41 42An Examplecustomer1 2 3 4 Which movie is best? y= X +movie 1 2 3 4 1 ? ? 3 5 ? ? 3 3 1 ? Can we guess ratings for customer/movie combinations not in the dataset? = + 4 1 3 5 3 3 11 1 1 1 1 1
Iowa State - STAT - 511
Estimating Estimable Functions of :An Example movie 1 2 3 4 1 ? ? 3 5 ? ? 3 3 1 ? customer i's rating of movie j + Ci + mj +ijcustomer1 2 3 4 =Can we guess ratings for customer/movie combinations not in the dataset? Which movie is best?YijYij=Cop
Iowa State - STAT - 511
Alternative ParameterizationsFor example yij i = 1, 2, 3 j = 1, 2Recall that the Gauss-Markov Linear Model simply says that E(y) C(X) and Var(y) = 2 I for some 2 &gt; 0.Treatment Effects E(yij ) = + i 1 2 3 Cell Means E(yij ) = iThus, as long as C(X) = C
Iowa State - STAT - 511
Inference Under the Normal Theory Gauss-Markov Linear Model Inference (cont.)Remember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) We've answered all questions except the last one: y 3) When does t = [(1 -
Iowa State - STAT - 511
Inference Under the Normal Theory Gauss-Markov Linear ModelRemember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) We've answered all questions except the last one: 3) When does t = [(1 - 2 ) - (1 - 2 )] / s2
Iowa State - STAT - 511
Practical Data AnalysisA not uncommon situation: A client brings you data from a food development study: 3 treatments:Old &quot;on market&quot; formulation of soup new formulation, Same salt content as old new formulation, Reduced salt contentThey recruited 25 p
Iowa State - STAT - 511
Practical Data AnalysisPractical Data AnalysisHow would you analyze the data? (N.B. all analyses account for blocking / pairing of obs. within subject) 1. ANOVA F test of O = S = R , report p-value and trt. means 2. 3 paired t-tests: O = S , O = R , and
Iowa State - STAT - 511
POWER OF THE F-TESTA very common consulting question: &quot;I'm planning a study to do .&quot;. How many replicates (per treatment) should I use? I know 5 ways that can be used to determine an appropriate sample sizeAs many as you can afford (time, money) n = 3 p
Iowa State - STAT - 511
POWER OF THE F-TESTSuppose C is a q p matrix such that C is testable. Earlier, we established that the quadratic form incorporating ^ C - d has a non-central F distribution ^ F = (C - d) Fq,n-k where 2 = (C - d) [C(X X)- C ] 2-1 ( 2 )A very common cons
Iowa State - STAT - 511
REDUCED vs. FULL MODEL F-TESTTests of C - d = 0 lead to F tests, but that's not the only way to an F test 500/402: big emphasis on model comparison SS in ANOVA tables usually explained in terms of model comparison e.g. SS for AB interaction is the differ
Iowa State - STAT - 511
REDUCED vs. FULL MODEL F-TESTWhy does model comparison lead to F tests? If you test Ho using a C test or using a model comparison test, do you get the same answer? Again, will answer using a general setup (Normal GM model) y = X + , N(0, 2 I)Questions a
Iowa State - STAT - 511
Equivalence of model comparison and Cb F testsSummary of results from Cb estimates and testsContinuation of the Storage time example from Part 9. Data: Storage Temperature 20 C 30 C 2 5 Time Ho: 6 6 7 7 Temp Ho: 16 9 12 15 These correspond to tests of:
Iowa State - STAT - 511
Equivalence of model comparison and Cb F testsContinuation of the Storage time example from Part 9. Data: Storage Time 3 months 6 months Storage Temperature 20 C 30 C 2 5 6 6 7 7 9 12 15 16Copyright c 2011 Dept. of Statistics (Iowa State University)Sta
Iowa State - STAT - 511
ANalysis Of VAriance (ANOVA) for a sequence of models Some examplesMultiple RegressionModel comparison can be generalized to a sequence of models (not just one full and one reduced model) N(0, 2 I)X1 = 1, X2 = [1, x1 ], X3 = [1, x1 , x2 ], . . . Xm = [
Iowa State - STAT - 511
ANalysis Of VAriance (ANOVA) for a sequence of modelsModel comparison can be generalized to a sequence of models (not just one full and one reduced model) Context: usual nGM model: y = X + , Let X1 = 1 and Xm = X. But now, we have a sequence of models &quot;i
Iowa State - STAT - 511
THE AITKEN MODELAnalysis of averagesExamples - 1y = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I.V is assumed to be a known nonsingular Variance matrix.The Normal Theory Aitken Model adds an assu
Iowa State - STAT - 511
THE AITKEN MODELy = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I. V is assumed to be a known nonsingular Variance matrix. The Normal Theory Aitken Model adds an assumption of normality: N(0, 2 V) Obs
Iowa State - STAT - 511
the bootstrapProcess optimization Many physical processes can be described (at least approximately) as quadratic functions of input variable(s)ExamplesWe've seen a lot about inference on C in a nGM (or nAitken) modelA huge number of questions can be a
Iowa State - STAT - 511
the bootstrapWe've seen a lot about inference on C in a nGM (or nAitken) model A huge number of questions can be answered by appropriate choice of C key point is that C is a linear function of y What if quantity of interest is not a linear function of y?
Iowa State - STAT - 511
Randomization/permutation testsBootstrapping preserves the fixed effectsRandomization / Permutation testsResampling from a single pool of observations tests HoR : F1 (x) = F2 (x)Notice a subtle point: HoR is slightly more general than Ho:1 = 2H0R is
Iowa State - STAT - 511
Randomization / Permutation testsBootstrapping preserves the fixed effectsDifference of two means: resample Y1i and resample Y2i bootstrap estimates, 1B - 2B , are centered on/near Y1 - Y2 , ^ ^ which estimates 1 - 2 Regression bootstrap: ^ resample ^i
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSSeedling weight in 2 genotype study from Aitken model section. Seedling weight measured on each seedling. Two (potential) sources of variation: among flats and among seedlings within a flat. Yijk = + i + Tij + Tij ijk ijkExamp
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSStudies / data / models seen previously in 511 assumed a single source of &quot;error&quot; variation y = X + . are fixed constants (in the frequentist approach to inference) is the only random effect What if there are multiple sources of
Iowa State - STAT - 511
Experimental Designs and LME'sOne example:LME models provide one way to model correlations among observationsVery useful for experimental designs where there is more than one size of experimental unitOr designs where the observation unit is not the sa
Iowa State - STAT - 511
Experimental Designs and LME'sLME models provide one way to model correlations among observations Very useful for experimental designs where there is more than one size of experimental unit Or designs where the observation unit is not the same as the exp
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSThis is the commonly-used model for a CRD with t treatments, n experimental units per treatment, and m observations per experimental unit. We can write the model as y = X + Zu + , where X=[
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSA model for expt. data with subsampling yijk = + i + uij + eijk , (i = 1, ., t; j = 1, ., n; k = 1, ., m) = (, i , ., t ) , u = (u11 , u12 , ., utn ) , = (e111 , e112 , ., etnm ) , IRt+1 ,
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per blockijkYijk = + i + j + ij +where i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N.with ANOVA table:Expected Mean Squares from two different sources Source 1: Searle (19
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per block Yijk = + i + j + ij +ijkwhere i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N. with ANOVA table: Source Blocks Treatments BlockTrt Error C. total df B-1 T-1 (B-1)(T-1
Iowa State - STAT - 511
ANOVA ANALYSIS OF A BALANCED SPLIT-PLOT EXPERIMENTFieldBlock 1 0 100 150 50 150 100 50 100 150 0 50Plot Genotype B0Genotype CGenotype AExample: the corn genotype and fertilization response studyBlock 2 150 100 Block 3 100 50 0 0 150 50 0 0Main pl
Iowa State - STAT - 511
ANOVA ANALYSIS OF A BALANCED SPLIT-PLOT EXPERIMENTExample: the corn genotype and fertilization response study Main plots: genotypes, in blocks Split plots: fertilization 2 way factorial treatment structure split plot variability nested in main plot varia
Iowa State - STAT - 511
IDENTIFYING AN APPROPRIATE MODELGiven a description of a study, how do you construct an appropriate model?Context: more than one size of e.u.A made-up example, intended to be complicated (but far from being the most complicated I've seen)A study of th
Iowa State - STAT - 511
IDENTIFYING AN APPROPRIATE MODELGiven a description of a study, how do you construct an appropriate model? Context: more than one size of e.u. A made-up example, intended to be complicated (but far from being the most complicated I've seen) A study of th
Iowa State - STAT - 511
MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE GENERAL LINEAR MODELGiven a value of the parameter vector , f (w|) is a real-valued function of w.Suppose f (w|) is the probability density function (pdf ) or probability mass function (pmf ) of a random vec
Iowa State - STAT - 511
MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE GENERAL LINEAR MODELc 2011 Dept. Statistics (Iowa State University)Stat 511 section 211 / 23Suppose f (w|) is the probability density function (pdf ) or probability mass function (pmf ) of a random vector
Iowa State - STAT - 511
Prediction of random variablesKey distinction between fixed and random effects:Estimate means of fixed effects Estimate variance of random effectsBut in some instances, want to predict FUTURE values of a random effectExample (from Efron and Morris, 19